Problem 2 - Contemporary Precalculus

Question

In Exercises \(1-8,\) plot the point in the complex plane corresponding to the number. $$-7+6 i$$

Step-by-Step Solution

Answer

Answer: The coordinates of the point representing the complex number -7 + 6i in the complex plane are (-7, 6).

Step 1: Identify the real and imaginary parts

Identify the real part (x) and the imaginary part (y) of the complex number: -7 + 6i. Here, the real part x = -7 and the imaginary part y = 6.

Step 2: Plot the point in the complex plane

With the coordinates (-7, 6), plot the point in the complex plane: 1. Draw a horizontal axis (representing real numbers) and a vertical axis (representing imaginary numbers). 2. Locate the point (-7, 6) on the plane. 3. Put a dot at that point. The point (-7, 6) in the complex plane represents the complex number -7 + 6i.

Key Concepts

Complex Plane

The complex plane is a visual representation of complex numbers. Think of it like a graph where you can plot numbers that have both real and imaginary parts.

• The horizontal axis, also known as the real axis, represents the real part of the complex number.
• The vertical axis, called the imaginary axis, represents the imaginary part, usually indicated by the letter 'i'.

For complex numbers like \(-7 + 6i\), the point is plotted using these axes. By drawing both axes, you create a plane where each point corresponds to exactly one complex number. This allows you to visually explore how complex numbers relate to each other.

Real Part

The real part of a complex number is the component that doesn't involve the imaginary unit \('i'\). It behaves just like the regular numbers you are familiar with.

• In \(-7 + 6i\), the real part is \(-7\).
• This part tells us how far left or right to move along the real axis of the complex plane.

When plotting a complex number, you begin by moving horizontally based on its real part. So, for \(-7\), you would move 7 units to the left on the horizontal axis. It's crucial to identify the real part correctly to accurately place your point.

Imaginary Part

The imaginary part of a complex number is associated with the imaginary unit \('i'\). This part indicates how far to move up or down along the imaginary axis of the complex plane.

• For the complex number \(-7 + 6i\), the imaginary part is \(6 \, i\).
• This means you move 6 units up on the vertical axis, following the representation of the imaginary unit.

Understanding the imaginary part is essential for plotting points correctly. It complements the real part by completing the position of the complex number in the plane, offering a full picture of its location.